CoCalc Shared Fileswww / papers / motive_visibility / dsw_11.texOpen in CoCalc with one click!
Author: William A. Stein
1
2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3
%
4
% dsw.tex
5
%
6
% LAST MODIFIED: 12 December 2002
7
%
8
% AUTHORS: William Stein, Neil Dummigan, Mark Watkins
9
%
10
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
11
12
\documentclass{amsart}
13
\usepackage{amssymb}
14
\usepackage{amsmath}
15
\usepackage{amscd}
16
17
\newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}
18
19
\theoremstyle{plain}
20
\newtheorem{prop}{Proposition}[section]
21
\newtheorem{conj}[prop]{Conjecture}
22
\newtheorem{lem}[prop]{Lemma}
23
\newtheorem{thm}[prop]{Theorem}
24
\newtheorem{cor}[prop]{Corollary}
25
\theoremstyle{definition}
26
\newtheorem{defi}[prop]{Definition}
27
\theoremstyle{remark}
28
\newtheorem{examp}[prop]{Example}
29
\newtheorem{remar}[prop]{Remark}
30
31
\def\id{\mathop{\mathrm{ id}}\nolimits}
32
\DeclareMathOperator{\Ker}{\mathrm {Ker}}
33
\DeclareMathOperator{\Aut}{{\mathrm {Aut}}}
34
\renewcommand{\Im}{{\mathrm {Im}}}
35
\DeclareMathOperator{\ord}{ord}
36
\DeclareMathOperator{\End}{End}
37
\DeclareMathOperator{\Hom}{Hom}
38
\DeclareMathOperator{\Mor}{Mor}
39
\DeclareMathOperator{\Norm}{Norm}
40
\DeclareMathOperator{\Nm}{Nm}
41
\DeclareMathOperator{\tr}{tr}
42
\DeclareMathOperator{\Tor}{Tor}
43
\DeclareMathOperator{\Sym}{Sym}
44
\DeclareMathOperator{\Hol}{Hol}
45
\DeclareMathOperator{\vol}{vol}
46
\DeclareMathOperator{\tors}{tors}
47
\DeclareMathOperator{\cris}{cris}
48
\DeclareMathOperator{\length}{length}
49
\DeclareMathOperator{\dR}{dR}
50
\DeclareMathOperator{\lcm}{lcm}
51
\DeclareMathOperator{\Frob}{Frob}
52
\def\rank{\mathop{\mathrm{ rank}}\nolimits}
53
\newcommand{\Gal}{\mathrm {Gal}}
54
\newcommand{\Spec}{{\mathrm {Spec}}}
55
\newcommand{\Ext}{{\mathrm {Ext}}}
56
\newcommand{\res}{{\mathrm {res}}}
57
\newcommand{\Cor}{{\mathrm {Cor}}}
58
\newcommand{\AAA}{{\mathbb A}}
59
\newcommand{\CC}{{\mathbb C}}
60
\newcommand{\RR}{{\mathbb R}}
61
\newcommand{\QQ}{{\mathbb Q}}
62
\newcommand{\ZZ}{{\mathbb Z}}
63
\newcommand{\NN}{{\mathbb N}}
64
\newcommand{\EE}{{\mathbb E}}
65
\newcommand{\TT}{{\mathbb T}}
66
\newcommand{\HHH}{{\mathbb H}}
67
\newcommand{\pp}{{\mathfrak p}}
68
\newcommand{\qq}{{\mathfrak q}}
69
\newcommand{\FF}{{\mathbb F}}
70
\newcommand{\KK}{{\mathbb K}}
71
\newcommand{\GL}{\mathrm {GL}}
72
\newcommand{\SL}{\mathrm {SL}}
73
\newcommand{\Sp}{\mathrm {Sp}}
74
\newcommand{\Br}{\mathrm {Br}}
75
\newcommand{\Qbar}{\overline{\mathbb Q}}
76
\newcommand{\Xbar}{\overline{X}}
77
\newcommand{\Ebar}{\overline{E}}
78
\newcommand{\sbar}{\overline{s}}
79
\newcommand{\nf}[1]{\mbox{\bf #1}}
80
\newcommand{\fbar}{\overline{f}}
81
82
% ---- SHA ----
83
\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
84
\newcommand{\textcyr}[1]{%
85
{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
86
\selectfont #1}}
87
\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
88
89
\newcommand{\HH}{{\mathfrak H}}
90
\newcommand{\aaa}{{\mathfrak a}}
91
\newcommand{\bb}{{\mathfrak b}}
92
\newcommand{\dd}{{\mathfrak d}}
93
\newcommand{\ee}{{\mathbf e}}
94
\newcommand{\Fbar}{\overline{F}}
95
\newcommand{\CH}{\mathrm {CH}}
96
97
\begin{document}
98
\title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}
99
\author{Neil Dummigan}
100
\author{William Stein}
101
\author{Mark Watkins}
102
\date{12 December 2002}
103
\subjclass{11F33, 11F67, 11G40.}
104
105
\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
106
Shafarevich-Tate group.}
107
108
\address{University of Sheffield\\ Department of Pure
109
Mathematics\\
110
Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
111
U.K.}
112
\address{Harvard University\\Department of Mathematics\\
113
One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
114
\address{Penn State Mathematics Department\\
115
University Park\\State College, PA 16802\\ U.S.A.}
116
117
\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
118
\email{watkins@math.psu.edu}
119
120
121
\begin{abstract}
122
123
We study Shafarevich-Tate groups of motives attached to modular
124
forms on $\Gamma_0(N)$ of weight bigger than~$2$. We deduce a
125
criterion for the existence of nontrivial elements of these
126
Shafarevich-Tate groups, and give $16$ examples in which a strong
127
form of the Beilinson-Bloch conjecture implies the existence of
128
such elements. We also use modular symbols and observations about
129
Tamagawa numbers to compute nontrivial conjectural lower bounds on
130
the orders of the Shafarevich-Tate groups of modular motives of
131
low level and weight at most $12$. Our methods build upon the
132
idea of visibility due to Cremona and Mazur, but in the context of
133
motives instead of abelian varieties.
134
\end{abstract}
135
136
\maketitle
137
138
\section{Introduction}
139
Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
140
be the associated $L$-function. The conjecture of Birch and
141
Swinnerton-Dyer \cite{BSD} predicts that the order of vanishing of $L(E,s)$
142
at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
143
also gives an interpretation of the leading term in the Taylor
144
expansion in terms of various quantities, including the order of
145
the Shafarevich-Tate group of~$E$.
146
147
Cremona and Mazur \cite{CM} look, among all strong Weil elliptic
148
curves over $\QQ$ of conductor $N\leq 5500$, at those with
149
nontrivial Shafarevich-Tate group (according to the Birch and
150
Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
151
group has predicted elements of prime order~$p$. In most cases
152
they find another elliptic curve, often of the same conductor,
153
whose $p$-torsion is Galois-isomorphic to that of the first one,
154
and which has positive rank. The rational points on the second elliptic
155
curve produce classes in the common $H^1(\QQ,E[p])$. They show
156
\cite{CM2} that these lie in the Shafarevich-Tate group of the
157
first curve, so rational points on one curve explain elements of
158
the Shafarevich-Tate group of the other curve.
159
160
The Bloch-Kato conjecture \cite{BK} is the generalisation to
161
arbitrary motives of the leading term part of the Birch and
162
Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
163
\cite{B, Be} generalises the part about the order of vanishing at the
164
central point, identifying it with the rank of a certain Chow
165
group.
166
167
This paper is a partial generalisation of \cite{CM} and \cite{AS}
168
from abelian varieties over $\QQ$ associated to modular forms of
169
weight~$2$ to the motives attached to modular forms of higher weight.
170
It also does for congruences between modular forms of equal weight
171
what \cite{Du2} did for congruences between modular forms of different
172
weights.
173
174
We consider the situation where two newforms~$f$ and~$g$, both of
175
even weight $k>2$ and level~$N$, are congruent modulo a maximal
176
ideal $\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
177
$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
178
to order at least $2$ at $s=k/2$. In Section~\ref{sec:examples},
179
we give sixteen such examples (all with $k=4$ and $k=6$), and in
180
each example, we find that $\qq$ divides the numerator of the
181
algebraic number $L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$
182
is a certain canonical period.
183
184
In fact, we show how this divisibility may be deduced from the
185
vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The
186
point is, the congruence between$f$ and~$g$ leads to a congruence
187
between suitable ``algebraic parts'' of the special values
188
$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity
189
one result of Faltings and Jordan shows that the congruence of
190
Fourier expansions leads to a congruence of certain associated
191
cohomology classes. These are then identified with the modular
192
symbols which give rise to the algebraic parts of special values.
193
If $L(g,k/2)$ vanishes then the congruence implies that
194
$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.
195
196
The Bloch-Kato conjecture sometimes then implies that the
197
Shafarevich-Tate group $\Sha$ attached to~$f$ has nonzero
198
$\qq$-torsion. Under certain hypotheses and assumptions, the most
199
substantial of which is the Beilinson-Bloch conjecture relating
200
the vanishing of $L(g,k/2)$ to the existence of algebraic cycles,
201
we are able to construct some of the predicted elements of~$\Sha$
202
using the Galois-theoretic interpretation of the congruences to
203
transfer elements from a Selmer group for~$g$ to a Selmer group
204
for~$f$. One might say that algebraic cycles for one motive
205
explain elements of~$\Sha$ for the other, or that we use
206
congruences to link the Beilinson-Bloch conjecture for one motive
207
with the Bloch-Kato conjecture for the other.
208
%In proving the local
209
%conditions at primes dividing the level, and also in examining the
210
%local Tamagawa factors at these primes, we make use of a higher weight
211
%level-lowering result due to Jordan and Livn\'e \cite{JL}.
212
213
We also compute data which, assuming the Bloch-Kato conjecture,
214
provides lower bounds for the orders of numerous Shafarevich-Tate
215
groups (see Section~\ref{sec:invis}).
216
%Our data is consistent
217
%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the
218
%congruence modulus is necessarily a perfect square (assuming that~$\Sha$
219
%is finite).
220
221
\section{Motives and Galois representations}
222
This section and the next provide definitions of some of the
223
quantities appearing later in the Bloch-Kato conjecture. Let
224
$f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
225
$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,
226
which is necessarily totally real. Let~$\lambda$ be any finite
227
prime of~$E$, and let~$\ell$ denote its residue characteristic. A
228
theorem of Deligne \cite{De1} implies the existence of a
229
two-dimensional vector space $V_{\lambda}$ over $E_{\lambda}$, and
230
a continuous representation
231
$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
232
such that
233
\begin{enumerate}
234
\item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$
235
not dividing~$lN$, and
236
\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
237
characteristic polynomial of $\Frob_p^{-1}$ acting on
238
$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
239
\end{enumerate}
240
241
Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
242
the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
243
There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
244
both $2$-dimensional $E$-vector spaces. For details of the
245
construction see \cite{Sc}. The de Rham realisation has a Hodge
246
filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset
247
F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
248
cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
249
cohomology.
250
For each prime $\lambda$, there is a natural isomorphism
251
$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a
252
$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
253
each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
254
Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
255
There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
256
which amounts to multiplying the action of $\Frob_p$ by $p^j$.
257
258
Following \cite{BK} (Section 3), for $p\neq l$ (including
259
$p=\infty$) let
260
$$
261
H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
262
H^1(I_p,V_{\lambda}(j))).
263
$$
264
The subscript~$f$ stands for ``finite
265
part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
266
$I_p$ is the inertia subgroup, and the cohomology is for
267
continuous cocycles and coboundaries. For $p=l$ let
268
$$
269
H^1_f(\QQ_l,V_{\lambda}(j))=\ker
270
(H^1(D_l,V_{\lambda}(j))\rightarrow
271
H^1(D_l,V_{\lambda}(j)\otimes_{\QQ_l} B_{\cris}))
272
$$
273
(see Section 1 of
274
\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
275
$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
276
elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
277
in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
278
279
There is a natural exact sequence
280
$$
281
\begin{CD}0@>>>T_{\lambda}(j)@>>>
282
V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.
283
$$
284
Let
285
$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
286
Define the $\lambda$-Selmer group
287
$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
288
$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
289
$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
290
condition at $p=\infty$ is superfluous unless $l=2$. Define the
291
Shafarevich-Tate group
292
$$
293
\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/
294
\pi_*H^1_f(\QQ,V_{\lambda}(j)).
295
$$
296
Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
297
prime ideal~$\lambda$ is the length of the $\lambda$-component of
298
$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
299
and write~$\Sha$ for~$\Sha(k/2)$. It depends on the choice of
300
$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
301
each $V_{\lambda}$. But if $A[\lambda]$ is irreducible then
302
$T_{\lambda}$ is unique up to scaling and the $\lambda$-part of
303
$\Sha$ is independent of choices.
304
305
In the case $k=2$ the motive comes from an (self-dual)
306
isogeny class of abelian varieties over $\QQ$, with endomorphism
307
algebra containing $E$. If one chooses an abelian variety $B$ in
308
the isogeny class and takes all the $T_{\lambda}(1)$ to be
309
$\lambda$-adic Tate modules, then what we have defined above
310
coincides with the usual Shafarevich-Tate group of $B$. To see
311
this one uses 3.11 of \cite{BK}, for $l=p$. For $l\neq p$,
312
$H^1(\QQ_p,V_l)=0$. Considering the formal group, every class in
313
$B(\QQ_p)/lB(\QQ_p)$ is represented by an $l$-power torsion point
314
in $B(\QQ_p)$, so maps to zero in $H^1(\QQ_p,A_l)$.
315
316
Define the group of global torsion points
317
$$
318
\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).
319
$$
320
This is analogous to the group of rational torsion points on an
321
elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
322
which the exponent of any prime ideal~$\lambda$ is the length of
323
the $\lambda$-component of $\Gamma_{\QQ}$.
324
325
\section{Canonical periods}
326
We assume from now on for convenience that $N\geq 3$. We need to
327
choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
328
and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
329
in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$
330
agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$
331
and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in
332
\cite{DFG} using cohomology, with non-constant coefficients, of
333
modular curves. (In \cite{DFG}, see especially Sections 2.2 and
334
5.4, and the paragraph preceding Lemma 2.3.)
335
336
For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
337
module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
338
$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes
339
E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$-module
340
$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.
341
342
Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
343
generalised elliptic curves with full level-$N$ structure. Let
344
$\mathfrak{E}$ be the universal generalised elliptic curve over
345
$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
346
of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed
347
using a projector on the cohomology of a desingularisation of
348
$\mathfrak{E}^{k-2}$). Realising $M(N)(\CC)$ as the disjoint union
349
of $\phi(N)$ copies of the quotient
350
$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
351
completed upper half plane), and letting $\tau$ be a variable on
352
$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to
353
the elliptic curve with period lattice generated by $1$ and
354
$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on
355
the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.
356
Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$
357
is a well-defined differential form on (a desingularisation of)
358
$\mathfrak{E}^{k-2}$ and naturally represents a generating element
359
of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at
360
primes dividing $Nk!$ so that this is the case.) We shall call
361
this element $e(f)$.
362
363
Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and
364
$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
365
a natural action of complex conjugation on $V_B$, breaking it up
366
into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
367
Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
368
to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
369
$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
370
are rank one $O_E$-modules, but not necessarily free, since the
371
class number of $O_E$ may be greater than one. Choose nonzero
372
elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
373
the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
374
$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
375
376
\section{The Bloch-Kato conjecture}\label{sec:bkconj}
377
In this section we extract from the Bloch-Kato conjecture for
378
$L(f,k/2)$ a prediction about the order of the Shafarevich-Tate
379
group, by analysing the other terms in the formula.
380
381
Let $L(f,s)$ be the $L$-function attached to~$f$. For
382
$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
383
Euler product
384
$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
385
there is an analytic continuation given by an integral, as
386
described in the next section. Suppose that $L(f,k/2)\neq 0$. The
387
Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
388
following equality of fractional ideals of~$E$:
389
$$
390
\frac{L(f,k/2)}{\vol_{\infty}}=
391
\left(\prod_pc_p(k/2)\right)
392
\frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}.
393
$$
394
Here, {\bf and from this point onwards, }$\pm$ represents the
395
parity of $(k/2)-1$. The quantity
396
$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$
397
multiplied by the determinant of the isomorphism
398
$V_B^{\pm}\otimes\CC\simeq (V_{\dR}/F^{k/2})\otimes\CC$,
399
calculated with respect to the lattices $O_E\delta_f^{\pm}$ and
400
the image of $T_{\dR}$. For $l\neq p$, $\ord_{\lambda}(c_p(j))$ is
401
defined to be
402
\begin{align*}
403
\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
404
\ord_{\lambda}(P_p(p^{-j}))\\
405
=&\length\>\> \left(H^0(\QQ_p,A_{\lambda}(j))/H^0\left(\QQ_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p}\right)\right).
406
\end{align*}
407
408
We omit the definition of $\ord_{\lambda}(c_p(j))$ for
409
$\lambda\mid p$, which requires one to assume Fontaine's de Rham
410
conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
411
$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
412
concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$
413
is a prime of good reduction. For such primes, the de Rham
414
conjecture follows from Theorem 5.6 of \cite{Fa1}.)
415
416
Strictly speaking, the conjecture in \cite{BK} is only given for
417
$E=\QQ$. We have taken here the obvious generalisation of a slight
418
rearrangement of (5.15.1) of \cite{BK}. The Bloch-Kato conjecture
419
has been reformulated and generalised by Fontaine and Perrin-Riou,
420
who work with general $E$, though that is not really the point of
421
their work. In Section 11 of \cite{Fo2} it is sketched how to
422
deduce the original conjecture from theirs, though only in the
423
case $E=\QQ$.
424
\begin{lem}\label{vol}
425
$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
426
$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
427
\end{lem}
428
\begin{proof}
429
We note that $\Omega_{\pm}$ is equal to the determinant of the
430
period map from $F^{k/2}V_{\dR}\otimes\CC$ to
431
$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we
432
used above in the definition of $\vol_{\infty}$ (c.f. the last
433
paragraph of 1.7 of \cite{De2}). We are using here natural
434
pairings. Recall that the index of $O_E\delta_f^{\pm}$ in
435
$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed
436
by noting that, locally away from primes dividing $Nk!$, the index
437
of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its
438
dual, both being equal to the ideal denoted~$\eta$ in \cite{DFG2}.
439
\end{proof}
440
\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
441
Then the fractional ideal $c_p(j)$ is supported at most on
442
divisors of~$p$.
443
\end{lem}
444
\begin{proof}
445
As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
446
$\ord_{\lambda}(c_p(j))$ is the length of the finite
447
$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
448
where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
449
trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
450
torsion free.
451
\end{proof}
452
453
\begin{lem}\label{local1}
454
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
455
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
456
$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
457
that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not
458
congruent modulo $\qq$ to any newform of weight~$k$, trivial
459
character, and level dividing $N/p$. Then $\ord_{\qq}(c_p(j))=0$
460
for all integers~$j$.
461
\end{lem}
462
\begin{proof}
463
There is a natural injective map from
464
$V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$ to $H^0(I_p,A[\qq](j))$
465
(i.e., $A[\qq](j)^{I_p}$).
466
Consideration of $\qq$-torsion shows that
467
$$
468
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))\geq \dim_{E_{\qq}}
469
H^0(I_p,V_{\qq}(j)).
470
$$ To prove the lemma it suffices to show that
471
$$
472
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),
473
$$
474
since this ensures that $H^0(I_p,A_{\qq}(j))=
475
V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$, hence that
476
$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p})$.
477
If the dimensions differ then, given that $f$ is not congruent
478
modulo $\qq$ to a newform of level dividing $N/p$, Proposition 2.2
479
of \cite{L} shows that we are in the situation covered by one of
480
the three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv
481
-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
482
unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
483
result that $N$ is the prime-to-$q$ part of the conductor of
484
$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
485
the condition $q>k$) implies the existence of a newform of weight
486
$k$, trivial character and level dividing $N/p$, congruent to~$g$
487
modulo $\qq$. This contradicts our hypotheses.
488
\end{proof}
489
490
\begin{remar}
491
For an example of what can be done when~$f$ is congruent to
492
a form of lower level, see the first example in Section~\ref{sec:other_ex}
493
below.
494
\end{remar}
495
496
\begin{lem}\label{at q}
497
If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
498
$\ord_{\qq}(c_q)=0$.
499
\end{lem}
500
\begin{proof}
501
It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
502
application, at the end of Section~2.2, of the results of
503
\cite{Fa1}) that $T_{\qq}$ is the
504
$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
505
module $T_{\dR}\otimes O_{\qq}$ by the functor they call
506
$\mathbb{V}$. (This property is part of the definition of an
507
$S$-integral premotivic structure given in Section~1.2 of
508
\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
509
of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
510
in Theorem~4.1 of \cite{BK} follows from the first paragraph of
511
2(h) of \cite{Fa1}.)
512
\end{proof}
513
514
\begin{lem}
515
If $A[\lambda]$ is an
516
irreducible representation of $\Gal(\Qbar/\QQ)$,
517
then
518
$$\ord_{\lambda}(\#\Gamma_{\QQ})=0.$$
519
\end{lem}
520
\begin{proof}
521
This follows trivially from the definition.
522
\end{proof}
523
524
Putting together the above lemmas we arrive at the following:
525
\begin{prop}\label{sha}
526
Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that
527
$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$,
528
where $\qq\mid q$. Assume the same hypotheses as in Lemma
529
\ref{local1} for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which
530
locally at $\qq$ are as in the previous section. If
531
$L(f,k/2)\aaa^{\pm}/\vol_{\infty}\neq 0$ then the Bloch-Kato
532
conjecture predicts that
533
$$
534
\ord_{\qq}(\#\Sha)=\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty}).
535
$$
536
\end{prop}
537
538
\section{Congruences of special values}
539
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
540
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
541
large enough to contain all the coefficients $a_n$ and $b_n$.
542
Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv
543
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Assume
544
that $A[\qq]$ is an irreducible representation of
545
$\Gal(\Qbar/\QQ)$, and that $q\nmid N\phi(N)k!$. Choose
546
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
547
$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
548
$T_B^{\pm}$ locally at $\qq$. Make two further assumptions:
549
$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
550
551
\begin{prop} \label{div}
552
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
553
\end{prop}
554
\begin{proof} This is based on some of the ideas used in Section 1 of
555
\cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V},
556
which presumably should refer to ``Condition 2''. Since
557
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
558
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
559
1=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
560
$$
561
\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).
562
$$
563
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
564
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
565
where the integral is taken along the positive imaginary axis,
566
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
567
Thus we are reduced
568
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
569
570
Let $\mathcal{D}_0$ be the group of divisors of degree zero
571
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
572
integer $r\geq 0$, let $P_r(R)$ be the additive group of
573
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
574
groups have a natural action of $\Gamma_1(N)$. Let
575
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
576
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
577
578
Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
579
the argument in 1.7 of \cite{V}, the cohomology class
580
$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in
581
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
582
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are
583
now dealing with cohomology over $X_1(N)$ rather than $M(N)$,
584
which is why we insist that $q\nmid \phi(N)$. It follows from the
585
last line of Section~4.2 of \cite{St} that, up to some small
586
factorials which do not matter locally at $\qq$,
587
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
588
(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
589
$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
590
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
591
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
592
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
593
show is divisible by $\qq$.
594
Similarly
595
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
596
(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
597
$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
598
Therefore it would suffice to show that, for some $\mu\in O_E$,
599
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
600
$\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that,
601
for some $\mu\in O_E$, the element
602
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
603
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
604
non-constant coefficients. This would be the case if
605
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
606
one-dimensional subspace upon reduction modulo~$\qq$. But this is
607
a consequence of Theorem 2.1(1) of \cite{FJ} (for which we need
608
the irreducibility of $A[\qq]$).
609
\end{proof}
610
\begin{remar}\label{sign}
611
The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
612
equal. They are determined by the eigenvalue of the
613
Atkin-Lehner involution~$W_N$,
614
which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
615
$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
616
characteristic coprime to $2N$. The common sign in the functional
617
equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
618
$W_N$ acting on~$f$ and~$g$.
619
\end{remar}
620
621
This is analogous to the remark at the end of Section~3 of \cite{CM},
622
which shows that if~$\qq$ has odd residue characteristic and
623
$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
624
at least two at $s=k/2$. Note that Maeda's conjecture
625
implies that there are no examples of~$g$ of
626
level one with positive sign in their functional equation such that
627
$L(g,k/2)=0$ (see \cite{CF}).
628
629
\section{Constructing elements of the Shafarevich-Tate group}
630
Let~$f$,~$g$ and $\qq$ be as in the first paragraph of the
631
previous section. In the previous section we showed how the
632
congruence between $f$ and $g$ relates the vanishing of $L(g,k/2)$
633
to the divisibility by $\qq$ of an ``algebraic part'' of
634
$L(f,k/2)$. Conjecturally the former is associated with the
635
existence of certain algebraic cycles (for $M_g$) while the latter
636
is associated with the existence of certain elements of the
637
Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this
638
section we show how the congruence, interpreted in terms of Galois
639
representations, provides a direct link between algebraic cycles
640
and the Shafarevich-Tate group.
641
642
For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
643
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
644
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
645
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
646
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
647
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
648
649
Recall that $L(g,k/2)=0$ and $L(f,k/2)\neq 0$. Since the sign in
650
the functional equation for $L(g,s)$ is positive (this follows
651
from $L(f,k/2)\neq 0$, see Remark \ref{sign}), the order of
652
vanishing of $L(g,s)$ at $s=k/2$ is at least $2$. According to the
653
Beilinson-Bloch conjecture \cite{B,Be}, the order of vanishing of
654
$L(g,s)$ at $s=k/2$ is the rank of the group
655
$\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational rational equivalence
656
classes of null-homologous, algebraic cycles of codimension $k/2$
657
on the motive $M_g$. (This generalises the part of the
658
Birch--Swinnerton-Dyer conjecture which says that for an elliptic
659
curve $E/\QQ$, the order of vanishing of $L(E,s)$ at $s=1$ is
660
equal to the rank of the Mordell-Weil group $E(\QQ)$.)
661
662
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
663
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
664
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
665
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
666
get (assuming also the Beilinson-Bloch conjecture) a subspace of
667
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
668
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
669
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
670
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
671
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
672
Sections~1 and~6.5 of \cite{Fo2}. We shall call it the ``strong''
673
Beilinson-Bloch conjecture.
674
675
Similarly, if $L(f,k/2)\neq 0$ then we expect that
676
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
677
coincides with the $\qq$-part of $\Sha$.
678
\begin{thm}\label{local}
679
Let $q\nmid N$ be a prime satisfying $q>k$. Let~$r$ be the dimension
680
of $H^1_f(\QQ,V'_{\qq}(k/2))$. Suppose that $A[\qq]$ is an irreducible
681
representation of $\Gal(\Qbar/\QQ)$ and that for no prime $p\mid N$ is
682
$f$ congruent modulo $\qq$ to a newform of weight~$k$, trivial
683
character and level dividing $N/p$. Suppose that, for all primes
684
$p\mid N$, $\,p\not\equiv -w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$
685
if $p^2\mid N$. (Here $w_p$ is the common eigenvalue of the
686
Atkin-Lehner involution $W_p$ acting on $f$ and $g$.) Then the
687
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
688
$\FF_{\qq}$-rank at least $r$.
689
\end{thm}
690
691
\begin{proof}
692
The theorem is trivially true if $r=0$, so we assume that $r>0$.
693
It follows easily from our hypothesis that the rank of the free
694
part of $H^1_f(\QQ,T'_{\qq}(k/2))$ is~$r$. The natural map from
695
$H^1_f(\QQ,T'_{\qq}(k/2))/\qq H^1_f(\QQ,T'_{\qq}(k/2))$ to
696
$H^1(\QQ,A'[\qq](k/2))$ is injective. Take a nonzero class $c$ in
697
the image, which has $\FF_{\qq}$-rank $r$. Choose $d\in
698
H^1_f(\QQ,T'_{\qq}(k/2))$ mapping to $c$. Consider the
699
$\Gal(\Qbar/\QQ)$-cohomology of the short exact sequence
700
$$
701
\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},
702
$$
703
where~$\pi$ is multiplication by a uniformising element of
704
$O_{\qq}$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial.
705
Hence $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
706
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
707
we get a nonzero, $\qq$-torsion class $\gamma\in
708
H^1(\QQ,A_{\qq}(k/2))$.
709
710
Our aim is to show that $\res_p(\gamma)\in
711
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
712
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
713
714
\vspace{1em}
715
\noindent{\bf Case (1)} $p\nmid qN$:
716
717
Consider the $I_p$-cohomology of the short exact sequence above.
718
Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
719
A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
720
$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
721
$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
722
follows from the fact that $d\in H^1_f(\QQ,T'_{\qq}(k/2))$ that
723
the image in $H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is
724
zero, hence that the restriction of~$c$ to
725
$H^1(I_p,A'[\qq](k/2))\simeq H^1(I_p,A[\qq](k/2))$ is zero. Hence
726
the restriction of $\gamma$ to $H^1(I_p,A_{\qq}(k/2))$ is also
727
zero. By line~3 of p.~125 of \cite{Fl2},
728
$H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just contained in)
729
the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$ to
730
$H^1(I_p,A_{\qq}(k/2))$, so we have shown that $\res_p(\gamma)\in
731
H^1_f(\QQ_p,A_{\qq}(k/2))$.
732
733
\vspace{1em}
734
\noindent{\bf Case (2)} $p\mid N$:
735
736
First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
737
It suffices to show that
738
$$\hspace{3.5em}
739
\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),
740
$$
741
since then the natural map from $H^0(I_p,V'_{\qq}(k/2))$ to
742
$H^0(I_p, A'_{\qq}(k/2))$ is surjective; this may be done as in
743
the proof of Lemma \ref{local1}. It follows as above that the
744
image of $c\in H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is
745
zero. Then $\res_p(c)$ comes from
746
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
747
order of this group is the same as the order of the group
748
$H^0(\QQ_p,A[\qq](k/2))$ (this is Lemma 1 of \cite{W}), which we
749
claim is trivial. By the work of Carayol \cite{Ca1}, the level $N$
750
is the conductor of $V_{\qq}(k/2)$, so $p\mid N$ implies that
751
$V_{\qq}(k/2)$ is ramified at $p$, hence $\dim
752
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
753
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
754
consider the case where this common dimension is $1$. The
755
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
756
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication
757
by~$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq})$. It
758
follows from Theor\'eme A of \cite{Ca1} that this is the same as
759
the Euler factor at $p$ of $L(f,s)$. By Theorems 3(ii) and 5 of
760
\cite{AL}, it then follows that $p^2\nmid N$ and
761
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
762
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
763
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$) as
764
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
765
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
766
$\res_p(\gamma)=0$ and certainly lies in
767
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
768
769
\vspace{1em}
770
\noindent{\bf Case (3)} $p=q$:
771
772
Since $q\nmid N$ is a prime of good reduction for the motive
773
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
774
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
775
$V'_{\qq}$ have the same dimension, where
776
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
777
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
778
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
779
the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
780
filtered module $T_{\dR}\otimes O_{\qq}$. Since also $q>k$, we may
781
now prove, in the same manner as Proposition 9.2 of \cite{Du3},
782
that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$. For the
783
convenience of the reader, we give some details.
784
785
In Lemma 4.4 of \cite{BK}, a cohomological functor $\{h^i\}_{i\geq
786
0}$ is constructed on the Fontaine-Lafaille category of filtered
787
Dieudonn\'e modules over $\ZZ_q$. $h^i(D)=0$ for all $i\geq2$ and
788
all $D$, and $h^i(D)=\Ext^i(1_{FD},D)$ for all $i$ and $D$, where
789
$1_{FD}$ is the ``unit'' filtered Dieudonn\'e module.
790
791
Now let $D=T_{\dR}\otimes O_{\qq}$ and $D'=T'_{\dR}\otimes
792
O_{\qq}$. By Lemma 4.5 (c) of \cite{BK},
793
$$
794
\hspace{3.5em} h^1(D)\simeq H^1_e(\QQ_q,T_{\qq}),
795
$$
796
where
797
$$
798
\hspace{3.5em}H^1_e(\QQ_q,T_{\qq})=\ker(H^1(\QQ_q,T_{\qq})\rightarrow
799
H^1(\QQ_q,V_{\qq})/H^1_e(\QQ_q,V_{\qq}))
800
$$
801
and
802
$$
803
\hspace{3.5em}H^1_e(\QQ_q,V_{\qq})=\ker(H^1(\QQ_q,V_{\qq})\rightarrow
804
H^1(\QQ_q,B_{\cris}^{f=1}\otimes_{\QQ_q} V_{\qq})).
805
$$ Likewise
806
$h^1(D')\simeq H^1_e(\QQ_q,T'_{\qq}).$ When applying results of
807
\cite{BK} we view $D$, $T_{\qq}$ etc. simply as $\ZZ_q$-modules,
808
forgetting the $O_{\qq}$-structure.
809
810
For an integer $j$ let $D(j)$ be $D$ with the Hodge filtration
811
shifted by $j$. Then
812
$$\hspace{3.5em}
813
h^1(D(j))\simeq H^1_e(\QQ_q,T_{\qq}(j))
814
$$
815
(as long as $k-p+1<j<p-1$, so that $D(j)$ satisfies the hypotheses
816
of Lemma 4.5 of \cite{BK}). By Corollary 3.8.4 of \cite{BK},
817
$$
818
\hspace{3.5em}
819
H^1_f(\QQ_q,V_{\qq}(j))/H^1_e(\QQ_q,V_{\qq}(j))\simeq
820
(D(j)\otimes_{\ZZ_q}\QQ_q)/(1-f)(D(j)\otimes_{\ZZ_q}\QQ_q),
821
$$
822
where $f$ is the Frobenius operator on crystalline cohomology. By
823
1.2.4(ii) of \cite{Sc}, and the Weil conjectures,
824
$H^1_e(\QQ_q,V_{\qq}(j))=H^1_f(\QQ_q,V_{\qq}(j))$, since $j\neq
825
(k-1)/2$. Similarly
826
$H^1_e(\QQ_q,V'_{\qq}(j))=H^1_f(\QQ_q,V'_{\qq}(j))$.
827
828
We have
829
$$\hspace{3.5em}h^1(D(k/2))\simeq H^1_f(\QQ_q,T_{\qq}(k/2))\quad\text{and}\quad
830
h^1(D'(k/2))\simeq H^1_f(\QQ_q,T'_{\qq}(k/2)).$$
831
The exact sequence in the middle of page 366 of \cite{BK} gives us a
832
commutative diagram.
833
$$\hspace{3.5em}\begin{CD}
834
h^1(D'(k/2))@>\pi >>h^1(D'(k/2))@>>>h^1(D'(k/2)/\qq D'(k/2))\\
835
@VVV@VVV@VVV\\
836
H^1(\QQ_q,T'_{\qq}(k/2))@>\pi
837
>>H^1(\QQ_q,T'_{\qq}(k/2))@>>>H^1(\QQ_q,A'[\qq](k/2)).
838
\end{CD}$$
839
The vertical arrows are all inclusions and we know that the image
840
of $h^1(D'(k/2))$ in $H^1(\QQ_q,T'_{\qq}(k/2))$ is exactly
841
$H^1_f(\QQ_q,T'_{\qq}(k/2))$. The top right horizontal map is
842
surjective since $h^2(D'(k/2))=0$.
843
844
The class $\res_q(c)\in H^1(\QQ_q,A'[\qq](k/2))$ is in the image
845
of $H^1_f(\QQ_q,T'_{\qq}(k/2))$, by construction, and therefore is
846
in the image of $h^1(D'(k/2)/\qq D'(k/2))$. By the fullness and
847
exactness of the Fontaine-Lafaille functor \cite{FL} (see Theorem
848
4.3 of \cite{BK}), $D'(k/2)/\qq D'(k/2)$ is isomorphic to
849
$D(k/2)/\qq D(k/2)$.
850
851
It follows that the class $\res_q(c)\in H^1(\QQ_q,A[\qq](k/2))$ is
852
in the image of $h^1(D(k/2)/\qq D(k/2))$ by the vertical map in
853
the exact sequence analogous to the above. Since the map from
854
$h^1(D(k/2))$ to $h^1(D(k/2)/\qq D(k/2))$ is surjective,
855
$\res_q(c)$ lies in the image of $H^1_f(\QQ_q,T_{\qq}(k/2))$. From
856
this it follows that $\res_q(\gamma)\in
857
H^1_f(\QQ_q,A_{\qq}(k/2))$, as desired.
858
\end{proof}
859
860
Theorem~2.7 of \cite{AS} is concerned with verifying local
861
conditions in the case $k=2$, where~$f$ and~$g$ are associated
862
with abelian varieties~$A$ and~$B$. (Their theorem also applies to
863
abelian varieties over number fields.) Our restriction outlawing
864
congruences modulo $\qq$ with cusp forms of lower level is
865
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
866
$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
867
curve with $\ord_l(j(A))<0$, consideration of a Tate
868
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
869
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
870
at~$l$.)
871
872
In this paper we have encountered two technical problems which we
873
dealt with in quite similar ways:
874
\begin{enumerate}
875
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
876
\item proving local conditions at primes $p\mid N$, for an element
877
of $\qq$-torsion.
878
\end{enumerate}
879
If our only interest was in testing the Bloch-Kato conjecture at
880
$\qq$, we could have made these problems cancel out, as in Lemma
881
8.11 of \cite{DFG}, by weakening the local conditions. However, we
882
have chosen not to do so, since we are also interested in the
883
Shafarevich-Tate group, and since the hypotheses we had to assume
884
are not particularly strong. Note that, since $A[\qq]$ is
885
irreducible, the $\qq$-part of $\Sha$ does not depend on the
886
choice of $T_{\qq}$.
887
888
\section{Examples and Experiments}
889
\label{sec:examples} This section contains tables and numerical
890
examples that illustrate the main themes of this paper. In
891
Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms},
892
which contains~$16$ examples of pairs $f,g$ such that the strong
893
Beilinson-Bloch conjecture and Theorem~\ref{local} together imply
894
the existence of nontrivial elements of the Shafarevich-Tate group
895
of the motive attached to~$f$. Section~\ref{sec:howdone} outlines
896
the higher-weight modular symbol computations that were used in
897
making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses
898
Table~\ref{tab:invisforms}, which summarizes the results of an
899
extensive computation of conjectural orders of Shafarevich-Tate
900
groups for modular motives of low level and weight.
901
Section~\ref{sec:other_ex} gives specific examples in which
902
various hypotheses fail. Note that in \S 7 ``modular symbol'' has
903
a different meaning from in \S 5, being related to homology rather
904
than cohomology. For precise definitions see \cite{SV}.
905
906
\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
907
\begin{table}
908
\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
909
910
$$
911
\begin{array}{|c|c|c|c|c|}\hline
912
g & \deg(g) & f & \deg(f) & q\text{'}s \\\hline
913
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
914
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
915
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
916
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
917
\vspace{-2ex} & & & & \\
918
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
919
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
920
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
921
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
922
\vspace{-2ex} & & & & \\
923
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
924
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
925
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
926
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
927
\vspace{-2ex} & & & & \\
928
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
929
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
930
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
931
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
932
\hline
933
\end{array}
934
$$
935
\end{table}
936
937
938
Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
939
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
940
along with at least one prime~$q$ such that there is a prime
941
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
942
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
943
The notation is as follows.
944
The first column contains a label whose structure is
945
\begin{center}
946
{\bf [Level]k[Weight][GaloisOrbit]}
947
\end{center}
948
This label determines a newform $g=\sum a_n q^n$, up to Galois
949
conjugacy. For example, \nf{127k4C} denotes a newform in the third
950
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
951
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
952
\ldots)$, then by the sequence of absolute values $|\mbox{\rm
953
Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
954
being first in the event that the two absolute values are equal,
955
and the first Galois orbit is denoted {\bf A}, the second {\bf B},
956
and so on. The second column contains the degree of the field
957
$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
958
contain~$f$ and its degree, respectively. The fifth column
959
contains at least one prime~$q$ such that there is a prime
960
$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
961
hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
962
satisfied for~$f$,~$g$, and~$\qq$.
963
964
For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
965
prime $q$ appears in the $q$-column, meaning $q^2$ divides the
966
order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
967
at the end of 7.3 below.
968
969
970
We describe the first line of Table~\ref{tab:newforms}
971
in more detail. See the next section for further details
972
on how the computations were performed.
973
974
Using modular symbols, we find that there is a newform
975
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
976
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
977
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
978
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
979
coefficients generate a number field~$K$ of degree~$17$, and by
980
computing the image of the modular symbol $XY\{0,\infty\}$ under
981
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
982
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
983
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
984
both equal to
985
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
986
+ \cdots\in \FF_{43}[[q]].$$
987
988
There is no form in the Eisenstein subspaces of
989
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
990
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
991
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
992
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
993
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
994
of Theorem~\ref{local}, so if $r$ is the dimension of
995
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
996
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
997
998
Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
999
$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
1000
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
1001
the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
1002
we have constructed the $\qq$-torsion in $\Sha$ predicted by the
1003
Bloch-Kato conjecture.
1004
1005
For particular examples of elliptic curves one can often find and
1006
write down rational points predicted by the Birch and
1007
Swinnerton-Dyer conjecture. It would be nice if likewise one could
1008
explicitly produce algebraic cycles predicted by the
1009
Beilinson-Bloch conjecture in the above examples. Since
1010
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
1011
0.3.2 of \cite{Z}), so ought to be trivial in
1012
$\CH_0^{k/2}(M_g)\otimes\QQ$.
1013
1014
\subsection{How the computation was performed}\label{sec:howdone}
1015
We give a brief summary of how the computation was performed. The
1016
algorithms that we used were implemented by the second author, and
1017
most are a standard part of MAGMA (see \cite{magma}).
1018
1019
Let~$g$,~$f$, and~$q$ be some data from a line of
1020
Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
1021
verified the existence of a congruence modulo~$q$, that
1022
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
1023
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
1024
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
1025
1026
To prove there is a congruence, we showed that the corresponding
1027
{\em integral} spaces of modular symbols satisfy an appropriate
1028
congruence, which forces the existence of a congruence on the
1029
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
1030
irreducible by computing a set that contains all possible residue
1031
characteristics of congruences between~$g$ and any Eisenstein
1032
series of level dividing~$N$, where by congruence, we mean a
1033
congruence for all Fourier coefficients of index~$n$ with
1034
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
1035
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
1036
listing a basis of such~$h$ and finding the possible congruences,
1037
where again we disregard the Fourier coefficients of index not
1038
coprime to~$N$.
1039
1040
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
1041
modular symbol ${\mathbf e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$
1042
under a map with the same kernel as the period mapping, and found that the
1043
image was~$0$. The period mapping sends the modular
1044
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
1045
so that ${\mathbf e}$ maps to~$0$ implies that
1046
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
1047
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
1048
=(-1)^{k/2} g$ which, because of the functional equation, implies
1049
that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
1050
independent interest because it includes examples of modular forms
1051
of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
1052
the functional equation. We found no such examples of weights
1053
$\geq 8$.
1054
1055
\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
1056
In this section we apply some of the results of
1057
Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
1058
of Shafarevich-Tate groups of many modular motives. The results of
1059
this section suggest that~$\Sha$ of a modular motive is usually not
1060
``visible at level~$N$'', i.e., explained by congruences at level~$N$,
1061
which agrees with the observations of \cite{CM} and \cite{AS}. For
1062
example, when $k>6$ we find many examples of conjecturally
1063
nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
1064
1065
For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d}
1066
L(f^{(i)},s)$ where $f^{(i)}$ runs over the
1067
$\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus
1068
$\CC^d/(2\pi i)^{k/2}\mathcal{L}$, where the lattice $\mathcal{L}$
1069
is defined by integrating integral cuspidal modular symbols (for
1070
$\Gamma_0(N)$) against the conjugates of~$f$. Let
1071
$\Omega_{M_f/\QQ}$ denote the volume of the $(-1)^{(k/2)-1}$
1072
eigenspace $T^{\pm}=\{z \in T : \overline{z}=(-1)^{(k/2)-1}z\}$
1073
for complex conjugation on~$T$.
1074
1075
1076
{\begin{table}
1077
\vspace{-2ex}
1078
\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
1079
\vspace{-4ex}
1080
1081
$$
1082
\begin{array}{|c|c|c|c|}\hline
1083
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
1084
\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\
1085
\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\
1086
\nf{263k4B} & 39 & 41^{2} & 263 \\
1087
\nf{269k4C} & 39 & 23^{2} & 269 \\
1088
\nf{271k4B} & 39 & 29^{2} & 271 \\
1089
\nf{281k4B} & 40 & 29^{2} & 281 \\
1090
\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\
1091
\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\
1092
%\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\
1093
% 319k4C removed since Lemma not satisfied.
1094
\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\
1095
\hline
1096
\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\
1097
\nf{101k6B} & 24 & 17^{2} & 101 \\
1098
\nf{103k6B} & 24 & 23^{2} & 103 \\
1099
\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\
1100
\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\
1101
\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\
1102
\nf{157k6B} & 34 & 251^{2} & 157 \\
1103
\nf{167k6B} & 40 & 41^{2} & 167 \\
1104
\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\
1105
\nf{173k6B} & 39 & 71^{2} & 173 \\
1106
\nf{181k6B} & 40 & 107^{2} & 181 \\
1107
\nf{191k6B} & 46 & 85091^{2} & 191 \\
1108
\nf{193k6B} & 41 & 31^{2} & 193 \\
1109
\nf{199k6B} & 46 & 200329^2 & 199 \\
1110
\hline
1111
\nf{47k8B} & 16 & 19^{2} & 47 \\
1112
\nf{59k8B} & 20 & 29^{2} & 59 \\
1113
\nf{67k8B} & 20 & 29^{2} & 67 \\
1114
\nf{71k8B} & 24 & 379^{2} & 71 \\
1115
\nf{73k8B} & 22 & 197^{2} & 73 \\
1116
\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\
1117
\nf{79k8B} & 25 & 307^{2} & 79 \\
1118
\nf{83k8B} & 27 & 1019^{2} & 83 \\
1119
\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\
1120
\nf{89k8B} & 29 & 44491^{2} & 89 \\
1121
\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\
1122
\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\
1123
\nf{103k8B} & 32 & 75367^{2} & 103 \\
1124
\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\
1125
\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\
1126
\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\
1127
\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\
1128
\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\
1129
\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\
1130
\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\
1131
\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\
1132
\hline
1133
\end{array}
1134
$$
1135
\end{table}
1136
\begin{table}
1137
$$
1138
\begin{array}{|c|c|c|c|}\hline
1139
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
1140
\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\
1141
\nf{121k8G} & 12 & 13^{2} & 3, 11 \\
1142
\nf{121k8H} & 12 & 19^{2} & 5, 11 \\
1143
\nf{125k8D} & 16 & 179^{2} & 5 \\
1144
\nf{127k8B} & 39 & 59^{2} & 127 \\
1145
\nf{128k8F} & 4 & 11^{2} & 1 \\
1146
\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\
1147
\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\
1148
\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\
1149
\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\
1150
\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\
1151
\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\
1152
\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397,
1153
639259, 17440535
1154
97287 \\
1155
\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241,
1156
769, 36583
1157
\\
1158
\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577,
1159
518737, 9837145
1160
699 \\
1161
\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\
1162
\nf{148k8B} & 11 & 19^{2} & 3, 37 \\
1163
\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\
1164
1165
\hline
1166
1167
\nf{43k10B} & 17 & 449^{2} & 43 \\
1168
\nf{47k10B} & 20 & 2213^{2} & 47 \\
1169
\nf{53k10B} & 21 & 673^{2} & 53 \\
1170
\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\
1171
\nf{59k10B} & 25 & 37^{2} & 59 \\
1172
\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\
1173
\nf{64k10K} & 2 & 19^{2} & 3 \\
1174
\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\
1175
\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\
1176
\nf{71k10B} & 30 & 1103^{2} & 71 \\
1177
1178
\hline
1179
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\
1180
\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\
1181
\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\
1182
\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\
1183
\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\
1184
\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\
1185
\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\
1186
\nf{49k12H} & 12 & 271^{2} & 7 \\
1187
\hline
1188
\end{array}
1189
$$
1190
\end{table}
1191
1192
\begin{lem}\label{lem:lrat}
1193
If $p\nmid Nk!$ is such that $f$ is not congruent to any of its
1194
Galois conjugates modulo a prime dividing $p$ then the $p$-parts
1195
of
1196
$$
1197
\frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad
1198
\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)
1199
$$
1200
are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
1201
\end{lem}
1202
1203
\begin{proof} (Sketch.) Let~$H$ be the $\ZZ$-module of all
1204
integral cuspidal modular symbols for $\Gamma_0(N)$. Let~$I$ be the
1205
image of~$H$ under the projection into $H\otimes\QQ$ corresponding
1206
to~$f$ and its Galois conjugates. Note that~$I$ is not necessarily
1207
contained in~$H$ since we will have inverted the residue
1208
characteristics of any primes of congruence between~$f$ and cuspforms
1209
of weight~$k$ for $\Gamma_0(N)$ which are not Galois conjugate to~$f$.
1210
1211
Now $\mathcal{L}$ is (up to divisors of $Nk!$) the lattice
1212
obtained by pairing the cohomology modular symbols
1213
$\Phi_{f^{(i)}}^{\pm}$ (as in \S 5) with the homology modular
1214
symbols in $H$, or equivalently in $I$. For $1\leq i\leq d$ let
1215
$I_i$ be the $O_E$-module generated by the image of the projection
1216
of $I$ into $I\otimes E$ corresponding to $f^{(i)}$. The finite
1217
index of $I\otimes O_E$ in $\oplus_{i=1}^d I_i$ is divisible only
1218
by primes of congruence between $f$ and its Galois conjugates. Up
1219
to these primes, $\Omega_{M_f/\QQ}/(2\pi i)^{((k/2)-1)d}$ is then
1220
a product of the $d$ complex numbers obtained by pairing
1221
$\Phi_{f^{(i)}}^{\pm}$ with $I_i$, for $1\leq i\leq d$. Bearing in
1222
mind the last line of \S 3, and ignoring divisors of $\aaa^{\pm}$,
1223
which are clearly of no importance, we see that these complex
1224
numbers are the $\Omega^{\pm}_{f^{(i)}}$, up to divisors of $Nk!$.
1225
We have then a factorisation of the left hand side which shows it
1226
to be equal to the right hand side, to the extent claimed by the
1227
lemma.
1228
\end{proof}
1229
1230
\begin{remar}
1231
The newform $f=\nf{319k4C}$ is congruent to one of its Galois conjugates
1232
modulo~$17$ and $17\mid \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}$ so the lemma
1233
and our computations
1234
say nothing about whether or not $17$ divides
1235
$\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)$.
1236
\end{remar}
1237
1238
1239
Let~$\mathcal{S}$ be the set of newforms with~level $N$ and
1240
weight~$k$ satisfying either $k=4$ and $N\leq 321$, or $k=6$ and
1241
$N\leq 199$, or $k=8$ and $N\leq 149$, or $k=10$ and $N\leq 72$,
1242
or $k=12$ and $N\leq 49$. Given $f\in \mathcal{S}$, let~$B$ be
1243
defined as follows:
1244
\begin{enumerate}
1245
\item Let $L_1$ be the numerator of the
1246
rational number $L(M_f/\QQ,k/2)/\Omega_{M_f/\QQ}$.
1247
If $L_1=0$ let $B=1$ and terminate.
1248
\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
1249
\item Let $L_3$ be the part of $L_2$ that is coprime to
1250
$p+1$ for every prime~$p$ such that $p^2\mid N$.
1251
\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
1252
of any prime of
1253
congruence between~$f$ and a form of weight~$k$ and
1254
lower level. (By congruence here, we mean a congruence for coefficients
1255
$a_n$ with $n$ coprime to the level of~$f$.)
1256
\item Let $L_5$ be the part of $L_4$ coprime to the residue characteristic
1257
of any prime of congruence
1258
between~$f$ and an Eisenstein series. (This eliminates
1259
residue characteristics of reducible representations.)
1260
\item Let $B$ be the part of $L_5$ coprime to the residue characteristic
1261
of any prime of congruence between $f$ and any one of its Galois
1262
conjugates.
1263
\end{enumerate}
1264
Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if
1265
$\ord_p(B) > 0$ then, according
1266
to the Bloch-Kato conjecture, $\ord_p(\#\Sha)=\ord_p(B) > 0$.
1267
1268
We computed~$B$ for every newform in~$\mathcal{S}$. There are
1269
many examples in which $L_3$ is large, but~$B$ is not, and this is
1270
because of Tamagawa factors. For example, {\bf 39k4C} has
1271
$L_3=19$, but $B=1$ because of a $19$-congruence with a form of
1272
level~$13$; in this case we must have $19\mid c_{3}(2)$, where
1273
$c_{3}(2)$ is as in Section~\ref{sec:bkconj}. See
1274
Section~\ref{sec:other_ex} for more details. Also note that in
1275
every example~$B$ is a perfect square, which is as predicted by
1276
the existence of Flach's generalised Cassels-Tate pairing
1277
\cite{Fl2}. (Note that for $\lambda\mid l$, a non-congruence prime
1278
for $f$, the lattice $T_{\lambda}$ is self-dual, so the pairing
1279
shows that the order of the $\ell$-part of $\Sha$, if finite,
1280
is a square.) That our computed value of~$B$ should be a square is
1281
not {\it a priori} obvious.
1282
1283
For simplicity, we discard residue characteristics instead of primes
1284
of rings of integers, so our definition of~$B$ is overly conservative.
1285
For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
1286
in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
1287
some prime above~$5$, but the prime of congruences of
1288
characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
1289
Eisenstein.
1290
1291
1292
The newforms for which $B>1$ are given in
1293
Table~\ref{tab:invisforms}. The second column of the table records the
1294
degree of the field generated by the Fourier coefficients of~$f$. The
1295
third contains~$B$. Let~$W$ be the intersection of the span of all
1296
conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and $W^{\perp}$ the
1297
Petersson orthogonal complement of~$W$ in $S_k(\Gamma_0(N),\ZZ)$. The
1298
fourth column contains the odd prime divisors of
1299
$\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which are exactly the
1300
possible primes of congruence between~$f$ and non-conjugate cusp forms
1301
of the same weight and level. We place a~$*$ next to the four entries
1302
of Table~\ref{tab:invisforms} that also occur in
1303
Table~\ref{tab:newforms}.
1304
1305
\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
1306
We have some other examples where forms of
1307
different levels are congruent.
1308
However, Remark~\ref{sign} does not
1309
apply, so that one of the forms could have an odd functional
1310
equation, and the other could have an even functional equation.
1311
For instance, we have a $19$-congruence between the
1312
newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier
1313
coefficients coprime to $39$.
1314
Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$
1315
has {\it odd} functional equation.
1316
Here~$f$ fails the condition about not being congruent
1317
to a form of lower level, so in Lemma~\ref{local1} it is possible that
1318
$\ord_{\qq}(c_{3}(2))>0$. In fact this does happen. Because
1319
$V'_{\qq}$ (attached to~$g$ of level $13$) is unramified at $p=3$,
1320
$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1321
two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1322
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1323
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1324
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1325
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1326
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1327
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1328
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1329
our example here with $p=3$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1330
nontrivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1331
does not work. This is just as well, since had it
1332
worked we would have expected
1333
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1334
shows not to be the case.
1335
1336
In the following example, the divisibility between the levels is
1337
the other way round. There is a $7$-congruence between
1338
$g=\nf{122k6A}$ and $f=\nf{61k6B}$, both $L$-functions have even
1339
functional equation, and $L(g,3)=0$. In the proof of
1340
Theorem~\ref{local}, there is a problem with the local condition
1341
at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
1342
$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
1343
kernel is at most one dimensional, so we still get the
1344
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1345
$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
1346
elements of $\Sha$ for \nf{61k6B} (assuming all along the strong
1347
Beilinson-Bloch conjecture). In particular, these elements of
1348
$\Sha$ are {\it invisible} at level 61. When the levels are
1349
different we are no longer able to apply Theorem 2.1 of \cite{FJ}.
1350
However, we still have the congruences of integral modular symbols
1351
required to make the proof of Proposition \ref{div} go through.
1352
Indeed, as noted above, the congruences of modular forms were
1353
found by producing congruences of modular symbols. Despite these
1354
congruences of modular symbols, Remark~\ref{sign} does not apply,
1355
since there is no reason to suppose that $w_N=w_{N'}$, where $N$
1356
and $N'$ are the distinct levels.
1357
1358
Finally, there are two examples where we have a form $g$ with even
1359
functional equation such that $L(g,k/2)=0$, and a congruent form
1360
$f$ which has odd functional equation; these are a 23-congruence
1361
between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1362
between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1363
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1364
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1365
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1366
the appropriate modification of Theorem \ref{local} (with strong
1367
Beilinson-Bloch conjecture) does not necessarily provide us with
1368
nontrivial $\qq$-torsion in $\Sha$. It only tells us that the
1369
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ has
1370
$\FF_{\qq}$-rank at least $1$. It could all be in the image of
1371
$H^1_f(\QQ,V_{\qq}(2))$. $\Sha$ appears in the conjectural formula
1372
for the first derivative of the complex $L$ function, evaluated at
1373
$s=k/2$, but in combination with a regulator that we have no way
1374
of calculating.
1375
1376
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1377
associated with $f$ and $g$ by the construction of Mazur, Tate and
1378
Teitelbaum \cite{MTT}, each divided by a suitable canonical
1379
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1380
quite clear what to make of this. This divisibility may be proved
1381
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1382
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1383
normalised) are congruent $\bmod{\,\qq}$, as a result of the
1384
congruence between the modular symbols out of which they are
1385
constructed. Integrating an appropriate function against these
1386
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1387
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1388
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1389
where the signs in the functional equations of $L(g,s)$ and
1390
$L_q(g,s)$ are the same, positive in this instance. (According to
1391
the proposition in Section 18 of \cite{MTT}, the signs differ
1392
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
1393
1394
We also found some examples for which the conditions of
1395
Theorem~\ref{local} were not met. For example, we have a
1396
$7$-congruence between \nf{639k4B} and \nf{639k4H}, but
1397
$w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a
1398
similar problem with a $7$-congruence between \nf{260k6A} and
1399
\nf{260k6E} --- here $w_{13}=1$ so that $13\equiv
1400
-w_{13}\pmod{7}$. According to Propositions \ref{div} and
1401
\ref{sha}, Bloch-Kato still predicts that the $\qq$-part of $\Sha$
1402
is non-trivial in these examples. Finally, there is a
1403
$5$-congruence between \nf{116k6A} and \nf{116k6D}, but here the
1404
prime~$5$ is less than the weight~$6$ so Propositions \ref{div}
1405
and \ref{sha} (and even Lemma~\ref{lem:lrat}) do not apply.
1406
1407
\begin{thebibliography}{AL}
1408
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
1409
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
1410
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
1411
Shafarevich-Tate groups of abelian varieties, preprint.
1412
\bibitem[BS-D]{BSD}
1413
B. J. Birch, H. P. F. Swinnerton-Dyer,
1414
Notes on elliptic curves. I and II.
1415
{\em J. reine angew. Math. }{\bf 212 }(1963), 7--25,
1416
{\bf 218 }(1965), 79--108.
1417
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
1418
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
1419
\bibitem[BCP]{magma}
1420
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
1421
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
1422
235--265, Computational algebra and number theory (London, 1993).
1423
\bibitem[Be]{Be} A. Beilinson, Height pairing between algebraic cycles,
1424
{\em in }Current trends in arithmetical algebraic
1425
geometry (K. Ribet, ed.) {\em Contemp. Math. }{\bf 67 }(1987), 1--24.
1426
\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
1427
of motives, The Grothendieck Festschrift Volume I, 333--400,
1428
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
1429
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
1430
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
1431
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
1432
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
1433
Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
1434
Duke Math. J. }{\bf 59 }(1989), 785--801.
1435
\bibitem[CM1]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
1436
Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
1437
13--28.
1438
\bibitem[CM2]{CM2} J. E. Cremona, B. Mazur, Appendix to A. Agashe,
1439
W. Stein, Visible evidence for the Birch and Swinnerton-Dyer
1440
conjecture for modular abelian varieties of rank zero, preprint.
1441
\bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of
1442
$L_f(s)$ at the center of the critical strip, preprint.
1443
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
1444
$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
1445
{\bf 179}, 139--172, Springer, 1969.
1446
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
1447
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
1448
part 2, 313--346.
1449
\bibitem[DFG1]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
1450
of modular forms and the Tamagawa number conjecture, preprint.
1451
{{\sf
1452
http://www.andromeda.rutgers.edu/\~{\mbox{}}liguo/lgpapers.html}}
1453
\bibitem[DFG2]{DFG2} F. Diamond, M. Flach, L. Guo, The Bloch-Kato
1454
conjecture for adjoint motives of modular forms, {\em Math. Res.
1455
Lett. }{\bf 8 }(2001), 437--442.
1456
\bibitem[Du1]{Du3} N. Dummigan, Symmetric square $L$-functions and
1457
Shafarevich-Tate groups, {\em Experiment. Math. }{\bf 10 }(2001),
1458
383--400.
1459
\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
1460
Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
1461
\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
1462
Galois representations, {\em in }Algebraic analysis, geometry and
1463
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
1464
Press, Baltimore, 1989.
1465
\bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology
1466
and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
1467
\bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate
1468
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
1469
\bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations,
1470
S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
1471
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
1472
Boston Berlin, 1993.
1473
\bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de
1474
repr\'esentations $p$-adiques du groupe de Galois d'un corps
1475
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
1476
}{\bf 115 }(1982), 529--577.
1477
\bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des
1478
fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em
1479
Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249.
1480
\bibitem[FL]{FL} J.-M. Fontaine, G. Lafaille, Construction de
1481
repr\'esentations $p$-adiques, {\em Ann. Sci. E.N.S. }{\bf 15
1482
}(1982), 547--608.
1483
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon''
1484
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
1485
51--56.
1486
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
1487
representations coming from modular forms, {\em J. Number Theory
1488
}{\bf 31 }(1989), 133--141.
1489
\bibitem[MTT]{MTT} B. Mazur, J. Tate, J. Teitelbaum, On $p$-adic
1490
analogues of the conjectures of Birch and Swinnerton-Dyer, {\em
1491
Invent. Math. }{\bf 84 }(1986), 1--48.
1492
\bibitem[Ne]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
1493
heights. The arithmetic and geometry of algebraic cycles (Banff,
1494
AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
1495
Soc., Providence, RI, 2000.
1496
\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
1497
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
1498
\bibitem[SV]{SV} W. A. Stein, H. A. Verrill, Cuspidal modular
1499
symbols are transportable, {\em L.M.S. Journal of Computational
1500
Mathematics }{\bf 4 }(2001), 170--181.
1501
\bibitem[St]{St} G. Stevens, $\Lambda$-adic modular forms of
1502
half-integral weight and a $\Lambda$-adic Shintani lifting.
1503
Arithmetic geometry (Tempe, AZ, 1993), 129--151, Contemp. Math.,
1504
{\bf 174}, Amer. Math. Soc., Providence, RI, 1994.
1505
\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
1506
formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
1507
\bibitem[W]{W} L. C. Washington, Galois cohomology, {\em in
1508
}Modular Forms and Fermat's Last Theorem, (G. Cornell, J. H.
1509
Silverman, G. Stevens, eds.),101--120, Springer-Verlag, New York,
1510
1997.
1511
\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
1512
{\em Invent. Math. }{\bf 130 }(1997), 99--152.
1513
\end{thebibliography}
1514
1515
1516
\end{document}
1517